arXiv:1006.5612v2 [math.CO] 3 Mar 2011
RATIONAL EHRHART QUASI-POLYNOMIALS
EVA LINKE
Abstract. Ehrhart’s famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral
dilation factor. We study the case of rational dilation factors. It turns
out that the number of integral points can still be written as a rational
quasi-polynomial. Furthermore, the coefficients of this rational quasipolynomial are piecewise polynomial functions and related to each other
by derivation.
1. Introduction
Let Rn be the n-dimensional Euclidean space and let Zn be the integral
lattice. For a set M ⊂ Rn , we denote by int(M ) its interior, by conv(M )
its convex hull, by aff(M ) its affine hull, by vol(M ) its volume, which is
the usual Lebesgue measure of M , and by dim(M ) its dimension, which is
defined as the dimension of its affine hull. By voldim(M ) (M ) we denote the
dim(M )-dimensional volume of M . A polytope is called integral (rational),
if all its vertices have integral (rational) coordinates. For a rational polytope
P , we denote by d(P ) the denominator of P , that is, the smallest number
k ∈ Z>0 such that kP is an integral polytope. In other words d(P ) is
the lowest common multiple of the denominators of all coordinates of the
vertices of P . Furthermore, let the i-index di (P ) of a rational polytope P
be the smallest number k ∈ Z≥0 such that for each i-face F of P the affine
space k aff(F ) contains integral points.
A function p : Z → Z is called a quasi-polynomial with period dP
if there exist
periodic functions pi : Z≥0 → Z with period d such that p(k) = ni=0 pi (k)ki .
Ehrhart’s Theorem states the following:
Theorem 1.1 (Ehrhart, 1962, [9], McMullen, 1978, [12]). Let P ⊂ Rn be a
rational polytope. Then
dim(P )
G(P, k) := #(kP ∩ Zn ) =
X
Gi (P, k)ki for k ∈ Z≥0 .
i=0
G(P, ·) : Z≥0 → Z≥0 is called the Ehrhart quasi-polynomial of P . For every
i, the coefficient Gi (P, k) depends only on the congruence class of k modulo
di (P ).
Supported by the Deutsche Forschungsgemeinschaft within the project He 2272/4-1.
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2
EVA LINKE
Here, Gi (P, ·) : Z≥0 → Q is a periodic function and di (P ) is a period of
Gi (P, ·). The second part of this statement is due to McMullen. Furthermore, G0 (P, 0) = 1, and Gdim(P ) (P, k) = voldim(P ) (P ) for all k ∈ Z≥0 such
that k aff(P ) contains integer points.
For an introduction into Ehrhart theory we refer to Beck and Robins [5].
Unfortunately, di (P ) is not necessarily the minimal period of Gi (P, ·), that
is, there is possibly a smaller integer number p < di (P ) such that p is a
period of Gi (P, ·). This phenomenon is called period collapse and has been
subject to active research in the last years. McAllister and Woods [11] studied the 1- and 2-dimensional case with the result that period collapse does
not occur in dimension 1, and they gave a characterization of those rational
polygons in dimension 2 whose Ehrhart quasi-polynomial is a polynomial.
They also showed that the minimal periods of Gi (P, ·) are not necessarily
decreasing with i. Woods [15] gave a polynomial-time algorithm in fixed
dimension which decides whether a given integer p is a period of all Gi (P, ·).
In [6], Beck, Sam and Woods constructed polytopes with no period collapse at all. Furthermore, they showed that period collapse never occurs for
Gdim(P )−1 (P, ·). Haase and McAllister [10] gave a conjectural explanation
of period collapse involving splitting the polytope into pieces and applying
unimodular transformations onto these pieces.
We show that the Ehrhart quasi-polynomial can be generalized to a rational quasi-polynomial by allowing rational dilation factors, where a rational quasi-polynomial
with period d is a function p : Q → Q of the form
Pn
i
p(r) = i=0 pi (r)r where pi : Q → Q is periodic with period d.
Theorem 1.2. Let P ⊂ Rn be a rational polytope. Then for r ∈ Q≥0
dim(P )
Q(P, r) := #(rP ∩ Zn ) =
X
Qi (P, r)r i .
i=0
Here, Qi (P, ·) : Q≥0 → Q is a periodic function, and di (P ) is a period of
Qi (P, ·). We call Q(P, ·) : Q≥0 → Q the rational Ehrhart quasi-polynomial
of P .
We remark that Qi (P, ·) is an extension of Gi (P, ·) to rational numbers.
Hence, Q0 (P, 0) = G0 (P, 0) = 1. Furthermore, we have that Qdim(P ) (P, r) =
voldim(P ) (P ) for all r such that r aff(P ) contains integer points.
We define a rational analogue of the i-index. These rational indices allow us
to show a refined result on the periods of rational Ehrhart quasi-polynomials.
Definition 1.3. Let the rational denominator q(P ) of P be the smallest
positive rational number r such that rP is an integral polytope:
q(P ) = min{r ∈ Q>0 : rP is an integral polytope},
and let the rational i-index qi (P ) of P be the smallest positive rational number r such that for each i-face F of P the affine space r aff(F ) contains
RATIONAL EHRHART QUASI-POLYNOMIALS
3
integral points:
qi (P ) = min{r ∈ Q>0 : r aff(F ) ∩ Zn 6= ∅, for all i-faces F }.
Then we get the following result:
Corollary 1.4. Let P be a rational polytope. Then qi (P ) is a period of
Qi (P, ·). Furthermore, Q(P, ·) : Q≥0 → Z is a rational quasi-polynomial
with period q0 (P ) = q(P ).
Ehrhart’s reciprocity law is also true for rational Ehrhart quasi-polynomials:
Pdim(P )
Corollary 1.5. Let P be a rational polytope and let Q(P, r) = i=0 Qi (P, r)r i
be its rational Ehrhart quasi-polynomial. Then for r ∈ Q≥0 ,
#(r int(P ) ∩ Zn ) = (−1)dim(P ) Q(P, −r).
We show further that the coefficients Qi (P, ·) of the rational Ehrhart quasipolynomial are piecewise polynomials. Here, we assume P to be full-dimensional. This assumption is necessary since, if P is contained in an affine
hyperplane not containing 0, we have #(rP ∩ Zm ) = 0 for all r ∈ Q≥0 such
that r aff(P ) does not contain integral points. Thus, in that case Qi (P, r) =
0 for nearly all points r ∈ Q≥0 . On the other hand, if P is contained in a
hyperplane containing 0, it behaves like a full-dimensional polytope.
Theorem P
1.6. Let P ⊂ Rn be an n-dimensional rational polytope and let
Q(P, r) = ni=0 Qi (P, r)r i be its rational Ehrhart quasi-polynomial. Then
Qi (P, ·) is a piecewise polynomial of degree n − i.
By Q′i (P, r) we denote the first derivative of Qi (P, r) in r if it exists. Using
this we deduce the following relation between the coefficients of the rational
Ehrhart quasi-polynomial of a polytope P :
Theorem 1.7. Let P ⊂ Rn be an n-dimensional rational polytope. Then
Q′i (P, r) = −(i + 1)Qi+1 (P, r),
i = 0, . . . , n − 1,
for all r ≥ 0 where the derivative exists.
This theorem can be seen as a first step towards investigations on period
collapses; it implies that, in contrast to the integral case, the minimal periods
of Qi (P, r) are decreasing with i if 0 ∈ P .
In general nothing is known about extremal values of Gi (P, k) or Qi (P, r).
As a first result in this direction we have in the 2-dimensional case:
Theorem 1.8. Let P be an arbitrary 2-dimensional rational polygon. Then
|Q1 (P, r)| ≤ Q1 (P, 0) for all 0 ≤ r < q(P ).
To this end, we work out an explicit example using the approach presented
by Sam and Woods in [13]. An analogous statement for Q0 (P, r) is not true.
The paper is organized as follows: In Section 2 we present all tools used
for the proofs of the results presented in this introduction. The proofs of
Theorems 1.2, 1.6 and 1.7 and their Corollaries are given in this section
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EVA LINKE
as well. In Section 3 we work out a detailed example (see Theorem 3.1)
and deduce Theorem 1.8 from the explicit formulas of the rational Ehrhart
quasi-polynomial of this example.
For further information on Ehrhart (quasi-)polynomials and similar problems as considered in this work, we refer to [2, 4, 7, 8].
2. Rational dilations
To prove Theorem 1.2, we need a well-known property of the Gi (·, ·), which
we present here:
Pdim(P )
Lemma 2.1. Let G(P, k) = i=0 Gi (P, k)ki be the Ehrhart quasi-polynomial
of a rational polytope P . Then Gi (mP, k) = Gi (P, mk)mi for m, k ∈ Z≥0 .
Pdim(P )
Proof. We have #(mkP ∩ Zn ) = G(mP, k) =
Gi (mP, k)ki and
i=0
P
dim(P
)
#(mkP ∩ Zn ) = G(P, mk) =
Gi (P, mk)(mk)i . Comparing coefi=0
ficients yields Gi (mP, k) = Gi (P, mk)mi , ∀m, k ∈ Z≥0 (see Barvinok [3,
Section 4.3] for details on equality of quasi-polynomials).
Pdim(P )
Proof of Theorem 1.2. Let G(P, k) = i=0 Gi (P, k)ki for k ∈ Z≥0 be the
Ehrhart quasi-polynomial of P . We define
a
1
:= Gi
P, a bi .
Qi P,
b
b
a
ka
1
ka
Qi P, ab is well-defined,
since
for
=
we
get
Q
= Gi ( kb
P, ka)ki bi =
P,
i
b
kb
kb
Gi ( 1b P, a)bi = Qi P, ab by Lemma 2.1. Then
dim(P
dim(P )
a
a a i
X ) 1
X
1
i
Q P,
=G
P, a =
P, a a =
.
Gi
Qi P,
b
b
b
b
b
i=0
i=0
a
It remains to show thatQi P, b is periodic with period di (P ). Since b di (P )
is a multiple of di 1b P , we get
a
a
1
1
i
Qi P, + di (P ) = Gi
.
P, a + b di (P ) b = Gi
P, a bi = Qi P,
b
b
b
b
This proof implies that knowing the classical Ehrhart quasi-polynomial of
1
b P for all positive integers b is equivalent to knowing the rational Ehrhart
quasi-polynomial of P . However, as the next remark shows, it is not enough
to know the Ehrhart quasi-polynomial of a polytope to recover the rational
version:
Remark 2.2. Q(P, ·) : Q≥0 → Z is not invariant under translations of
P with respect to integral vectors. Furthermore, Q(P, ·) : Q≥0 → Z is not
RATIONAL EHRHART QUASI-POLYNOMIALS
5
necessarily monotonically increasing if 0 6∈ P . For instance, let
1/2
−1/2
0
T1 = conv
,
,
,
−1/2
−1/2
3/2
1/2
−1/2
0
T2 = conv
,
,
1/2
1/2
5/2
(see Figure 1).Then T2 = T1 + 01 . Nevertheless, we have Q(T1 , 2/3) = 2
and Q(T2 , 2/3) = 1. Moreover, Q(T2 , 2) = 7 and Q(T2 , 11/5) = 4.
2
2
1
-1
T1
T2
1
-1
1
-1
1
-1
Figure 1. Triangles T1 and T2
Furthermore, as for the integral case, there are examples of polytopes with
different combinatorial type that have the same rational Ehrhart quasipolynomial. Stanley [14] constructed two classes of polytopes, order polytopes and chain polytopes. In general these are polytopes of different combinatorial type but with the same Ehrhart polynomials. Since his consideration is independent of the integrality of dilation factors, these polytopes
have also the same rational Ehrhart polynomials.
Lemma 2.3. For all r, s ∈ Q we have Qi (sP, r) = Qi (P, sr)si .
Proof. Let r = ab , s = dc . By the definition of Qi (P, r) in the proof of
Theorem 1.2, we get, together with Lemma 2.1,
c a
c
Qi (sP, r) = Qi
= Gi
P,
P, a bi and
d b
db
c
ac ci
P,
a
bi .
=
G
Qi (P, sr)si = Qi P,
i
bd di
bd
As in the integral case, the rational indices are divisors of each other.
Lemma 2.4. Let P be a rational polytope. Then qi−1 (P )/qi (P ) ∈ Z for
i = 1, . . . , n.
i
Proof. Let H1i , . . . , Hm(i)
be the respective affine hulls of the m(i) i-faces of
i
P and let rj be the smallest positive rational number such that Hji contains
6
EVA LINKE
integral points. Then rHji contains integral points if and only if r is an
integral multiple of rji , for j = 1, . . . , m(j). Thus, qi (P ) is the smallest
positive rational number that is an integral multiple of all rji . Furthermore,
since Hji−1 ⊂ H̃i for some ̃, we have that rji−1 is an integral multiple of r̃i ,
and thus qi−1 (P ) is an integral multiple of qi (P ).
Now we are able to prove that the rational indices are periods of the coefficients of the rational Ehrhart quasi-polynomials.
Proof of Corollary 1.4. Since di (qi (P )P ) = 1 for all i, we know that
Qi (qi (P )P, r + k) = Qi (qi (P )P, r),
∀r ∈ Q≥0 , k ∈ Z≥0 .
This implies, together with Lemma 2.3,
Qi (P, r qi (P ) + k qi (P )) = Qi (P, r qi (P )),
∀r ∈ Q≥0 , k ∈ Z≥0 ,
and thus
Qi (P, r̃ + k qi (P )) = Qi (P, r̃),
∀r̃ ∈ Q≥0 , k ∈ Z≥0 .
Furthermore, together with Lemma 2.4, we get that q0 (P )/qi (P ) ∈ Z for
i = 1, . . . , n, and thus q0 (P ) = q(P ) is a period of Q(P, ·).
Concerning the minimal periods of Ehrhart quasi-polynomials, Beck, Sam
and Woods [6] showed that ddim(P )−1 (P ) is in fact the minimal period of
Gdim(P )−1 (P, ·). An analogous result is also true in the rational case:
Corollary 2.5. Let P ⊂ Rn be a rational polytope and let Q(P, r) =
Pdim(P )
Qi (P, r)r i be its rational Ehrhart quasi-polynomial. Then qdim(P )−1 (P )
i=0
is the minimal period of Qdim(P )−1 (P, ·).
Proof. By Lemma 2.3 and since qdim(P )−1 is homogeneous, it suffices to show
the statement for all P with qdim(P )−1 (P ) = 1. Thus we assume that st < 1
is a period of Qdim(P )−1 (P, ·) with s, t ∈ Z>0 , that is Qdim(P )−1 (P, r) =
Qdim(P )−1 P, r + st for all r ∈ Q≥0 . Again by Lemma 2.3 we get
1
1
Qdim(P )−1
P, rt = Qdim(P )−1
P, rt + s
for all r ∈ Q≥0 .
t
t
In particular this
is true for all rt ∈ Z≥0 , and thus, s is a period of
Gdim(P )−1 1t P, · , which is a contradiction, since
1
1
ddim(P )−1
P ≥ qdim(P )−1
P = t qdim(P )−1 (P ) = t > s.
t
t
We can also prove the Ehrhart reciprocity law for rational Ehrhart quasipolynomials. We refer to Beck and Robins [5, Chapter 4] for details on
Ehrhart reciprocity law.
RATIONAL EHRHART QUASI-POLYNOMIALS
7
Proof of Corollary 1.5. Let r = ab with a, b ∈ Z≥0 . Then, by Ehrhart reciprocity law and Lemma 2.3, we get
#
a
b
dim(P )
int(P ) ∩ Z
n
dim(P )
= (−1)
X
i=0
Gi
1
P, −a (−a)i
b
dim(P )
a 1 i
(−a)i
Qi P, −
= (−1)
b
b
i=0
a
dim(P )
.
= (−1)
Q P, −
b
dim(P )
X
For the proof of Theorem 1.6 we need the following Lemma:
Lemma 2.6. Let p : Q → Q be a rational quasi-polynomial of degree n ∈
Z>0 with period d ∈ Q>0 and constant leading coefficient, that is,
p(r) = pn r n + pn−1 (r)r n−1 + pn−2 (r)r n−2 + . . . + p1 (r)r + p0 (r),
where 0 6= pn ∈ Q and pi : Q → Q are periodic functions with period d for
i = 0, . . . , n − 1. Furthermore, suppose there exist an interval (r1 , r2 ) ⊂ Q
and ck ∈ Q for k ∈ Z≥0 such that
p(r + kd) = ck ,
∀r ∈ (r1 , r2 ), ∀k ∈ Z≥0 .
Then pi : (r1 , r2 ) → Q is a polynomial of degree n − i. Furthermore, if
pn > 0 then pn−1 has negative leading coefficient.
Proof. We prove this result by induction with respect to n. For n = 1 we
have c0 = p(r) = p1 r + p0 (r) for all r ∈ (r1 , r2 ). Thus, p0 (r) = c0 − p1 r for
r ∈ (r1 , r2 ), which is a polynomial of degree n − 0 = 1 with negative leading
coefficient.
Now let n > 1. We have
n
ck = pn · (r + kd) +
n−1
X
pi (r)(r + kd)i
∀r ∈ (r1 , r2 ), ∀k ∈ Z≥0 .
i=0
Pn−1
pi (r) (r + d)i − r i
Then q : Q → Q with q(r) := pn ·((r + d)n − r n )+ i=0
is a quasi-polynomial of degree n − 1 with period d and constant leading
coefficient, and
q(r + md) = pn · ((r + (m + 1)d)n − (r + md)n )
+
n−1
X
pi (r) (r + (m + 1)d)i − (r + md)i
i=0
= cm+1 − cm
∀r ∈ (r1 , r2 ), ∀m ∈ Z≥0 .
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EVA LINKE
Thus, we can use the induction hypothesis for q, and together with
q(r) = pn · ((r + d)n − r n ) +
n−1
X
pi (r) (r + d)i − r i
i=0


n−2
n−1
X
X
i
n
pn
= pn ndr n−1 +
pi (r)
di−j  r j
dn−j +
j
j
j=0
i=j+1
we get that
n−1
X
i i−j
n n−j
qj (r) := pn
pi (r)
d
d
+
j
j
i=j+1
is a polynomial of degree n−1−j for r ∈ (r1 , r2 ), for all j =
Since
0, . . . , n−2. n−1
d
pn nd > 0 we get, also by induction, that qn−2 (r) = pn n2 d2 + pn−1 (r) n−2
has a negative leading coefficient.
Now we use induction again to show that pj+1 is a polynomial of degree
n − j − 1 for r ∈ (r1 , r2 ). For j = n − 2 we have that qn−2 (r) = pn n2 d2 +
n−1
pn−1 (r) n−2
d is a polynomial of degree 1 with negative leading coefficient,
hence the same is true for pn−1 . For j < n − 2 write
n−j−1
n−1
X
X
i i−j
n n−j
qj (r) = pn
αi r i ∀r ∈ (r1 , r2 ).
pi (r)
d
=
d
+
j
j
i=0
i=j+1
Then for r ∈ (r1 , r2 ),
pj+1 (r)(j + 1)d =
n−j−1
X
i=0
i
αi r −
n−1
X
i=j+2
i i−j
n n−j
pi (r)
d − pn
d
j
j
which is a polynomial of degree n−j −1 since pi (r) is a polynomial of degree
n − i for i ≥ j + 2 by induction hypothesis.
We conclude that pi (r) is a polynomial of degree n − i for r ∈ (r1 , r2 )
and i = 1 . . . , n − 1. That p0 (r) is a polynomial follows immediately from
Pn−1
pi (r)r i .
p0 (r) = c0 − pn r n − i=1
Next we show that Ehrhart quasi-polynomials of full-dimensional rational
polytopes satisfy the setting in Lemma 2.6.
Proof of Theorem 1.6. By Theorem 1.2, Q(P, r) is a rational quasi-polynomial
of degree n with period q(P ) and constant, nonzero leading coefficient. To
apply Lemma 2.6 it remains to show that there exist 0 = r0 < r1 < . . . <
rl = q(P ) such that Q(P, r) is constant for r ∈ (ri + kq(P ), ri+1 + kq(P ))
and i = 0, . . . , l − 1, k ∈ Z≥0 . To this end we consider Q(P, r) as a function
Q≥0 → Q. Q(P, r) is certainly piecewise constant, and it jumps whenever
integral points leave or enter rP , which can only happen if one of the facets
of rP lies in a hyperplane containing integral points. Thus, for every facet F
of P let αF be the smallest positive rational number such that αF F lies in a
hyperplane containing integral points. Then {kαF : F facet of P, k ∈ Z≥0 }
RATIONAL EHRHART QUASI-POLYNOMIALS
9
are the only possible jump discontinuities of Q(P, r). By the definition of
q(P ), for a facet F of P there exists a kF ∈ Z such that kF αF = q(P ).
Thus for {r0 , . . . , rl } = {kαF : k = 0, . . . , kF , F facet of P } we can apply
Lemma 2.6.
We refer to Figure 3 in Section 3 for a visualization of the Qi (P, ·).
Remark 2.7. The rational Ehrhart quasi-polynomials can be extended to
real quasi-polynomials Q(P, ·) : R≥0 → Z≥0 . To do that, for r0 ∈ R≥0 \ Q
let rj ∈ Q≥0 , j ≥ 1 with r0 = limj→∞ rj ∈ R. We define Qi (P, r0 ) :=
limj→∞ Qi (P, rj ). This limit exists since Qi (P, ·) is piecewise continuous
and, for j large enough, all rj are contained in the same continuous part of
Qi (P, ·). Then, since Q(P, ·) : R≥0 → Z≥0 only jumps for rational points,
we get
#(r0 P ∩ Zn ) = lim #(rj P ∩ Zn ) = lim
j→∞
=
n
X
i=0
j→∞
lim Qi (P, rj )rji =
j→∞
n
X
Qi (P, rj )rji
i=0
n
X
Qi (P, r0 )r0i .
i=0
Baldoni et al. [1] generalized this statement to intermediate sums of rational polytopes and gave an efficient algorithm to compute these real quasipolynomials.
Now we show that the coefficients Qi (P, ·) are derivatives of each other.
Proof of Theorem 1.7. Let 0 = r0 < r1 < . . . < rl = q(P ) be as in the proof
of Theorem 1.6, and for m = 1, . . . , l, k ∈ Z≥0 let
cm,k = Q(P, r) =
n
X
Qi (P, r)r i
for r ∈ (rm−1 + k q(P ), rm + k q(P )).
i=0
Since Qi (P, r) is a polynomial of degree n − i in r, we can write it as
Qi (P, r) =
n−i
X
Qi,j r j .
j=0
Since Qi (P, r) are periodic with period q(P ), we can write r = r̃ + k q(P )
with k ∈ Z≥0 and rm−1 ≤ r̃ < rm for some m = 1, . . . , l and get
cm,k =
n
X
Qi (P, r̃)(r̃ + k q(P ))i =
=
h=0 i=0 j=max(0,h−i)
Qi,j r̃ j (r̃ + k q(P ))i
i=0 j=0
i=0
min(h,n−i)
n
n X
X
X
n−i
n X
X
i
Qi,j (k q(P ))i−h+j r̃ h ,
h−j
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EVA LINKE
which is a constant polynomial in r̃. Thus, for h 6= 0,
n
X
min(h,n−i)
X
i=0 j=max(0,h−i)
i
Qi,j (k q(P ))i−h+j = 0
h−j
and therefore
Q(P, r̃ + k q(P )) =
n
X
i
Qi,0 (k q(P )) =
n
X
Qi,0 (r − r̃)i .
i=0
i=0
Expanding to the quasi-polynomial form yields
Q(P, r̃ + k q(P )) =
n
X
Qi,0
i=0
=
n
X
j=0
i X
i
j=0
n−j X
i=0
j
r j r̃ i−j (−1)i−j
!
i+j
Qi+j,0 (−1)i r̃ i r j .
j
This implies that for all r̃ ∈ (rm−1 , rm ), m = 1, . . . , l,
Qj (P, r̃) =
n−j X
i+j
i=0
j
Qi+j,0 (−1)i r̃ i
and the claim follows by differentiation.
3. Dimension 2
In what follows, we denote by ⌊.⌋ the floor function, that is, ⌊x⌋ is the
largest integer not greater than x, by ⌈.⌉ the ceiling function, that is, ⌈x⌉ is
the smallest integer not smaller than x, and by {.} the fractional part, that
is, {x} = x − ⌊x⌋. For the following calculations, we mention
nr nt thentfollowing
fact: Let
n,
m,
t,
r
∈
Z,
m
>
0
and
t
≡
r
mod
m.
Then
m = m − m
nt
nr
and nt
m = m + −m .
First, we consider 2-dimensional triangles of the form T = conv 00 , xy1 , xy2 ,
where x1 < x2 ∈ Q and y ∈ Q>0 .
s1 a s2 a 0
Theorem 3.1. Let T = conv
, t1a b , t2a b
with a, b, t1 , t2 ∈
0
b
b
Z>0 , s1 , s2 ∈ Z, st22 > st11 , and gcd(a, b) = gcd(s1 , t1 ) = gcd(s2 , t2 ) = 1. Then
RATIONAL EHRHART QUASI-POLYNOMIALS
11
2 a
1 a
, )
( as
, ) ( as
bt2 b
bt1 b
a
b
( st11 , 1)
( st22 , 1)
Figure 2. Triangle T
for r ∈ Q≥0 the following hold:
1 a2 s 2 s 1
−
(i) Q2 (T, r) =
2 b2 t 2
t1
n o
1
ar
s2 s1
a t1 + t2
−
−
−
(ii) Q1 (T, r) =
b
2t1 t2
b
2
t2
t1
n
o
n
s2 s1
1 ar o2 s2 s1
1 ar
−
+2 +
−
(iii) Q0 (T, r) = 1 −
2 b
t2
t1
2 b
t2
t1
ar
t2 − 1 t1 − 1
+
+
lcm (t1 , t2 )
blcm (t1 , t2 )
2t2
2t1
−
{⌊ arb ⌋/lcm(tX
1 ,t2 )}lcm(t1 ,t2 ) i=0
s2 i
s1 i
s1 i
s2 i
−
+
−
.
t2
t2
t1
t1
P⌊at/b⌋
Proof. In what follows,
we determine Q(T, t) = i=0 Q(Q, i), t ∈ Q≥0 ,
s1
s2
t1
Q = conv
, t2
, (see Figure 2, [13]). For abbreviation we write
1
1
l instead of lcm (t1 , t2 ) and r for an arbitrary integer number with r ≡ t
mod lb. It is
s2 t
s1 t
2
−
+1
Q(Q, t) = #(tQ ∩ Z ) =
t2
t1
s2 r
s1 r
s2 s1
−
t−
+ −
+ 1.
=
t2
t1
t2
t1
This implies
(3.1)
Q(T, t) =
⌊at/b⌋ X
i=0
s2 s1
−
t2
t1
i−
s2 i
t2
s1 i
+ 1−
+ 1.
t1
12
EVA LINKE
Since ⌊at/b⌋ =
at
b
−
⌊at/b⌋
ar b , the first part can be written as
a2
a
+t
2b2
b
1 n ar o
1 n ar o2 1 n ar o
.
−
−
+
2
b
2 b
2 b
i=0
n o
For the second part, we remark that st22i is periodic with period t2 and
X
i = t2
l−1 X
s2 i
t2
i=0
t2 −1 l X
l(t2 − 1)
s2 i
.
=
=
t2
t2
2t2
i=0
Thus, we get
⌊at/b⌋ X
i=0
s2 i
t2
=
=
$ % l−1 at
X s2 i
b
l
i=0
t2
+
/l l
{⌊ ar
b ⌋ } X
i=0
s2 i
t2
/l l
{⌊ ar
b ⌋ } X
at n ar o l(t2 − 1)
s2 i
−
+
,
bl
bl
2t2
t2
i=0
and similarly
⌊at/b⌋ s1 i
−
t1
X
i=0
=
/l l
{⌊ ar
b ⌋ } X
at n ar o l(t1 − 1)
s1 i
−
+
.
1−
bl
bl
2t1
t1
i=0
After some elementary algebra, (3.1) expands to the claim.
In particular, the theorem shows that (as shown in Section 2) ab is a period
of Q1 (T, ·), Q1 (T, ·) is piecewise linear, and that blcm(ta 1 ,t2 ) is a period of
Q0 (T, r). Furthermore, Q0 (T, r) is piecewise quadratic, and the pieces are
equal up to a constant depending only on k (see Figure 3).
1
0
1
b
a
b
0
b
a
2b
a
lcm(t1 ,t2 )b
a
Figure 3. Continuous Q1 (P, r) and Q0 (P, r)
a si 0
Corollary 3.2. Let T be as in Theorem 3.1, gi = conv
,
, b ati
0
b
for i = 1, 2. Then |Q1 (T, r)| ≤ Q1 (T, 0) for all 0 ≤ r < blcm (t1 , t2 ). More
precisely, Q1 (T, r) ≥ −Q1 (T, 0) + Q1 (g1 , r) + Q1 (g2 , r).
RATIONAL EHRHART QUASI-POLYNOMIALS
13
j k
ar
ar
−
Proof. It is Q2 (gi , r) = 0, Q1 (gi , r) = btai , and Q0 (gi , r) = 1 − bt
bti
i
for all r ∈ Q, i = 1, 2. Since st22 − st11 > 0 we get
n o
1
ar
s2 s1
a t1 + t2
−
−
−
Q1 (T, r) =
b
2t1 t2
b
2
t2
t1
a t1 + t2 1 s 2 s 1
≤
+
−
= Q1 (T, 0).
b
2t1 t2
2 t2
t1
Furthermore
1 s2 s1
a t1 + t2 n ar o s2 s1
−
−
−
+
Q1 (T, r) =
b
2t1 t2
b
t2
t1
2 t2
t1
a t1 + t2
t1 + t2 1 s 2 s 1
a
−
−
+
−
≥
b
2t1 t2
2 t2
t1
b
t1 t2
= −Q1 (T, 0) + Q1 (g1 , r) + Q1 (g2 , r).
Remark 3.3. An analogous statement of Corollary 3.2 for Q0 (T, ·) is not
true. To see this, we consider the triangles
α−1 α+1 0
α
α
Tα = conv
,
,
, α ∈ Z.
0
1
1
Together with Theorem 3.1, we get, for m ∈ Z≥0 , 0 ≤ k < α and 0 ≤ r̃ < 1,
that
1
k(α − k − 2) + r̃ 2 − 2r̃ + 1.
Q0 (Tα , mα + k + r̃) =
α
Then for k ∼ α/2,
Q0 (Tα , mα + k + r̃) ∼
α
1 2
1
α
+
r̃ − 2r̃ ≥ −
4
α
4
α
which tends to infinity, when α goes to infinity, but Q0 (T, 0) = 1.
Now we can deduce the inequality |G1 (P, r)| ≤ G1 (P, 0) for arbitrary rational polygons:
Proof of Theorem 1.8. We first consider only integral dilation factors, that
is, we show that
(3.2)
|G1 (P, k)| ≤ G1 (P, 0)
for all k ∈ Z.
For this, we use several steps:
1. An integral version of Corollary 3.2 holds true for G1 (P, k) for arbitrary
triangles P with at least one integral vertex. This is true since G1 (P, k) is
invariant under translations and unimodular transformations.
2. (3.2) is true for every rational polygon P with one integral point in its
interior. Let P be an arbitrary 2-dimensional polygon with m vertices and
let z be an integral point in the interior of P . We consider the triangulation
of P as given in Figure 4.
14
EVA LINKE
g1
gm
T2
T1
Tm
g2
T3
z
g3
···
Figure 4. Triangulation of P
Here let g0 := gm . Then for all k ∈ Z,
m
m
X
X
G(gi , k) + 1.
G(Ti , k) −
G(P, k) =
i=1
i=1
Thus, expanding all Ehrhart polynomials yields
m X
1
1
G1 (P, k) =
G1 (Ti , k) − G1 (gi−1 , k) − G1 (gi , k) .
2
2
i=1
Thus, step 1 implies that G1 (P, k) ≤ G1 (P, 0), and
m X
1
1
G1 (Ti , k) − G1 (gi−1 , k) − G1 (gi , k)
G1 (P, k) =
2
2
i=1
m X
1
1
≥
−G1 (Ti , 0) + G1 (gi−1 , k) + G1 (gi , k)
2
2
i=1
= −G1 (P, 0).
3. (3.2) is true for arbitrary rational polygons P . Let l ∈ Z≥0 such that
(ld(P ) + 1)P contains at least one integral point in its interior. Then for all
k ∈ Z≥0 ,
G1 ((ld(P ) + 1)P, k) = G1 (P, k)(ld(P ) + 1).
Thus, from step c2 it follows that |Gi (P, k)| ≤ Gi (P, 0).
Finally we consider arbitrary rational dilation factors. Let P be an arbitrary
rational polygon and let r = pq ∈ Q≥0 with r ≤ d(P ), where p ∈ Z≥0 and
q ∈ Z>0 . Then, by Lemma 2.3, Qi (P, r) = Gi 1q P, p q i . Hence, (3.2)
implies that |Qi (P, r)| ≤ Qi (P, 0).
Acknowledgements
The author would like to thank Martin Henk for helpful discussions and
Matthias Beck and the reviewers for helpful comments improving the manuscript.
RATIONAL EHRHART QUASI-POLYNOMIALS
15
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Eva Linke, Fakultät für Mathematik, Universität Magdeburg, Universitätsplatz 2, D-39106-Magdeburg, Germany